Unstable shock formation of the Burgers-Hilbert equation

TL;DR

This paper establishes the existence of unstable shock solutions in the Burgers-Hilbert equation, demonstrating their formation from specific initial data and analyzing their blowup profiles with explicit parameters.

Contribution

It constructs smooth initial data leading to unstable shocks, proving their existence and detailed blowup behavior, which was previously conjectured.

Findings

Existence of unstable shock solutions proved.

Constructed initial data with finite $H^9$-norm leads to shocks.

Explicit blowup profile with cusp and H"older 1/5 continuity.

Abstract

This paper proves the existence of unstable shocks of the Burgers-Hilbert equation conjectured in arXiv:2006.05568. More precisely, we construct smooth initial data with finite $H^9$-norm such that the solution in self-similar coordinates is asymptotic to the first unstable solution to the self-similar inviscid Burgers equation. The blowup profile is a cusp with H\"older 1/5 continuity with explicit blowup time and location. Unlike the previously established stable shocks, the initial data cannot be taken in an open set; instead, we control the two unstable directions by Newton's iteration.